This is a bug fix to the control 018 implementation. The eddy phase speed presented in Bate's et al is really the doppler shifted phase speed. Previously the eddy phase speed (c) was limited to 20 cm/s here we limit the doppler shifted phase speed (u-c) to 20 cm/s
POP standard diagnostics for Exp 026
\(\color{green}{\quad CESM \quad Implementation \quad of \quad Steering \quad Parameterization}\)
During the development of the parameterization several tuning modifications were made to the terms listed above to better match the observations.
- New implementation of Eady Growth rate (sigma) that avoids undefined values at the equator. See alternate derivation at end of this page.
\(\quad\quad\quad \sigma_{vi} = {f \over \sqrt{R_i}}\) is replace by \(\sigma_{vi} = \quad {{\require\cancel \cancel f m^2} \over \cancel f N}\)
- Don't use Rhine Scale in calculation of \(L_{eddy}\)
\(\quad\quad\quad L_{eddy} = min (L_r, L_{req}, \cancel{L_{Rh}})\)
- Doppler Shifted Eddy phase speed calculation modified to avoid undefined value at the equator and limited to better match Bates paper.
\(\quad\quad\quad{c = - \beta * {L_r^2}}\)
\(\quad\quad\quad{|(u-c)|}\) limited to 20 cm/sec
- \(u_{rms}\) is hard to capture in this implememtation due to the coarse model resolution and use of mean vs resolved states. We imposed a \(5 cm/sec\) minimum on \(U_{rms}\) to tune to surface obs of Bates and included a scaling constant \(\alpha\) to handle the bias introduced by using mean state instead of resolved state.
\(\quad\quad\quad u_{rms} = max(5.,alpha*\sigma_{vi}*L_r)\)
\(\quad\quad\quad \alpha=4\) The scaling constant for \(U_{rms}\)
- The Eddy diffusivity calculated by our implementation was weak so we bumped up \(\Gamma\) from .35 to 1.75 (5x increase)
\(\quad\quad\quad \Gamma=.35\) replaced with \(\Gamma = 1.75\)
\(\color{green} {Parameterizing \quad the\quad Eddy\quad Length\quad Scale} \)
NOTES:
\(K=u_{rms}∗{\Gamma * \color{red}{L_{eddy}} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)}\)
\(L_{eddy} = min (L_r, L_{req}, \require{cancel}\cancel{L_{Rh}})\)
Since the two length scales we are using in the parameterization of \(L_{eddy}\) depend on the Baroclinic wave speed \(c_r\) we will first check \(c_r\) against Chelton 1998.
\(\quad\quad\quad\quad\quad\quad\quad\) First Baroclinic Wave Speed \(c_r\) CESM
\(\quad\quad\quad\quad\quad\quad\) Chelton Baroclinic gravity wave phase speed
\(\quad\quad\quad\) The barclinic wave speed looks reasonable so lets compare the Paramaterized Rossby Radius to Chelton
\(\quad\quad\quad\quad\quad\quad\) First Baroclinic Rossby Radius CESM \(\quad (L_{eddy})\)
\(\color{green} {Parameterizing \quad Zonal\quad Eddy\quad Phase\quad Speed}\quad (\color{red}{c})\)
NOTES:
\(K=u_{rms}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - \color{red}{c}|^2 /u_{rms}^2 (z=0)}\)
\(\require{cancel}\cancel{c = - \beta * {L_r^2}}\) \(L_r\) too high at equator
\(c = - \beta * L_{eddy}^2\)
\(\quad \quad \quad \)Eddy Phase speed CESM \((c = - \beta * L_{eddy}^2 )\)
(in control exp 018 this field was limited to 20cm/s not so in this run)
\(\quad\quad\quad\quad\quad\quad\) Hughes Phase Speed (cm/s) from Tulloch Marshall Smith '09
\(\color{green}{Zonal\quad Velocity \quad Term}\quad (\color{red}{u_{mean}})\)
NOTES:
\(K=u_{rms}∗{\Gamma * L_{eddy} \over (1 + b1 * |\color{red}{u_{mean}} - c|^2 /u_{rms}^2 (z=0))}\)
Here \(U_{mean}\) is just CESM surface velocity.
\(\quad\quad \quad \quad\quad \) Zonal Velocity CESM
\(\color{green}{(U-c)\quad Term}\)
NOTES:
- \((u-c)\) limited to 20cm/s
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\) (U-c) CESM
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\) (U-c) Bates et al
\(\color{green}{(U-c)^2 \quad Term}\)
NOTES:
- \(c = max(- \beta * L_{eddy}^2,-20)\)
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (U-c)^2\) CESM
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (U-c)^2\) Bates et al
\(\color{green}{Parameterizing \quad Eady \quad Growth \quad Rate}\quad \color{red}{\sigma}\)
NOTES:
- The original derivation of \(\sigma_{vi}\) goes to 0 at the equator because of \(f\)
\(\quad\quad\quad\quad\quad\sigma_{vi} = {f \over \sqrt{R_i}}\)
- The new derivation of \(\sigma_{vi}\):
\(\quad\quad\quad\quad\quad R_i = {f^2 N^2 \over {\underbrace{{ {g^2 \over \rho_0^2 } ( \frac{\partial \rho}{\partial y})^2 + {g^2 \over \rho_0^2 } ( \frac{\partial \rho}{\partial z})^2} }_\text{m^4}}}\quad\quad = \quad\quad {f^2N^2 \over m^4} \)
\(\quad\quad\quad\quad\quad \sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}\)
\(\quad\quad\quad\quad\quad \)Eady Growth Rate \((\sigma) \quad\) CESM
\(\quad\quad\quad\quad\quad \)Eady Growth Rate \((\sigma) \quad\) Bates et al
\(\color{green}{Parameterizing \quad RMS \quad eddy \quad velocity}\quad \color{red}{u_{rms}}\)
NOTES:
\(K=\color{red}{u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2(z=0)}\)
\(u_{rms} = alpha*{\sigma_{vi}}*L_{eddy}\)
- alpha (scaling constant) = 4
\(\sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}\)
\(u_{rms}\) is limited to 5 cm/s $ max(u_{rms},5.)$
\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Velocity \((u_{rms})\quad\) CESM
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad u_{rms} \) Bates et al
\(\color{green}{u_{rms}^2 \quad Term} \quad\)
NOTES:
- \(K={u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /\color{red}{u_{rms}^2} (z=0)} \)
- \(u_{rms}^2\) is a surface value
\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Velocity Squared \((u_{rms}^2)\quad\) CESM
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad u_{rms}^2 \) Bates et al
\(\color{green}{{(U-c)}^2 \over {u_{rms}^2}} \quad \color{green}{Term}\)
NOTES:
- \(K={u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * \color{red}{|u_{mean} - c|^2/{u_{rms}^2} (z=0)}} \)
- \(c = max(- \beta * L_{eddy}^2,-20)\)
- \(u_{rms} = max(u_{rms},5.)\)
- \(u_{rms}^2\) is a surface value
\(\quad\quad\quad\quad\quad\quad\) \({(U-c)}^2 \over {u_{rms}^2} \quad\) CESM
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {|u-c|^2 \over u_{rms}^2} \) Bates et al
\(\color{green}{\quad Suppression \quad factor = {1 \over (1 + b1 * |\bar u - c|^2 /u_{rms (z=0)}^2 )}}\)
NOTES:
- \(c = max(- \beta * L_{eddy}^2,-20)\)
- \(u_{rms} = max(u_{rms},5.)\)
- \(u_{rms}^2\) is a surface value
- \(b1\) (scaling constant) = 4.
\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor CESM
\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor Bates
\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor (Zonal x depth) CESM
\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor (Zonal x depth) Bates
\(\color{green} {\quad Parameterizing \quad The \quad Mixing \quad Term \quad }\color{red}{L_{mix}}\)
NOTES:
\(\color{red}{L_{mix}} = \Gamma * L_{eddy} * Suppression\)
\(Suppression= {1 \over (1 + b1 * |u_{mean} - c|^2 /u_{rms (z=0)}^2 )}\)
\(\color{red}{L_{mix}} = {\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)}\)
\(\Gamma = 1.75\) (Tuning mod: the original Gamma of .35 produced a Kappa with the correct structure but too weak.
\(\quad\quad\quad\quad\quad\quad\quad\) LMIX CESM
\(\color{green}{\quad Parameterizing \quad Eddy \quad Diffusivity \quad} (\color{red}{K})\)
NOTES:
- \(\color{red}{K}=u_{rms}∗L_{mix}\)
\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Diffusivity (K) CESM
\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Diffusivity (K) Bates
\(\quad\quad\quad\quad\quad\quad\quad\) (Zonal x Depth) CESM
\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\) K limited to (100 < K < 10000)
Here is the Bate's version
\(\quad\quad\quad\quad\quad\quad\quad\) Zonal average of the N2 normalized scaling CESM
The alternate derivation of \(\sigma_{vi}\):
\(R_i = {N^2\over{(\frac{\partial u}{\partial z})^2+(\frac{\partial v}{\partial z})^2}}\)
\(N^2={-g \over \rho_0 }\frac{\partial \rho}{\partial z}\)
After hydrostatic and geostrophic approximations
\(f \frac {\partial v}{\partial z} = {{-g \over \rho_0 }\frac {\partial \rho}{\partial x}}; \quad\quad f \frac {\partial u}{\partial z} = {{g \over \rho_0 }\frac {\partial \rho}{\partial y}} \)
so
\(\frac {\partial v}{\partial z} = {{-1\over f}{g \over \rho_0 }\frac {\partial \rho}{\partial x}}; \quad\quad \frac {\partial u}{\partial z} = {{1\over f}{g \over \rho_0 }\frac {\partial \rho}{\partial y}}\)
\(R_i = {f^2 N^2 \over {\underbrace{{ {g^2 \over \rho_0^2 } ( \frac{\partial \rho}{\partial y})^2 + {g^2 \over \rho_0^2 } ( \frac{\partial \rho}{\partial z})^2} }_\text{m^4}}}\quad\quad = \quad\quad {f^2N^2 \over m^4} \)
\(\sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}\)
\(RX_1 = RX_{east} = \Delta\rho_x = \rho_{i+1,j} - \rho_{i,j}\)
\(RY_1 = RY_{north} = \Delta\rho_y = \rho_{i,j+1} - \rho_{i,j}\)
\(RZ_1 = RZ_{k+1} = \Delta\rho_z = \rho_{k} - \rho_{k+1}\)
\(\displaystyle{1 \over L_{R_i}} \displaystyle\int_{2000m}^{100m} \left\lbrace { {-g\over\rho_0}{\frac {\partial \rho} {\partial z}} \over { {g^2 \over \rho_0^2 } \left[( \frac{\partial \rho}{\partial y})^2 + ( \frac{\partial \rho}{\partial z})^2\right] } } \right\rbrace dz\)
Note: missing \(f^2\) which will be cancelled when forming \(\sigma_{vi}\)
\(\quad\quad\) so \(\cdots\) this is not \(R_i\)
Implementation notes
Numerator : Top \(= -grav * RZ_{SAVE}(\cdots k+1) * dzwr(k)\)
Denominator :
\(\begin{align} work1 = p25 & * ( RX(..,i_{east},k)^2 \\ & + RX(..,i_{west},k)^2 \\ & + RX(..,i_{east},k+1)^2 \\ & + RX(..,i_{west},k+1)^2 ) / DXT(i,j)^2 \\ \end{align}\)
\(\begin{align} work2 = p25 & * ( RY(..,j_{north},k)^2 \\ & + RY(..,j_{south},k)^2 \\ & + RY(..,j_{north},k+1)^2 \\ & + RY(..,j_{south},k+1)^2 / DYT(i,j)^2 \\ \end{align}\)
\(\begin{align} work3 = {\left( TOP \over (grav^2*(work1+work2))\right)}*dzw(k) \end{align}\)
Notes:
1)Need to be careful at top and bottom of ocean
2)Accurate dzw(k) for each (i,j) to form $L_{R_i}$
3) When constructing $sigma$ itself, use $RZ_{SAVE}$ with a minimum N value
4) use eps2